• Keine Ergebnisse gefunden

Direct Proofs of the Existence of Equilibrium, the Gale-Nikaido-Debreu Lemma and the Fixed Point Theorems using Sperner’s Lemma

N/A
N/A
Protected

Academic year: 2022

Aktie "Direct Proofs of the Existence of Equilibrium, the Gale-Nikaido-Debreu Lemma and the Fixed Point Theorems using Sperner’s Lemma"

Copied!
24
0
0

Wird geladen.... (Jetzt Volltext ansehen)

Volltext

(1)

Munich Personal RePEc Archive

Direct Proofs of the Existence of

Equilibrium, the Gale-Nikaido-Debreu Lemma and the Fixed Point Theorems using Sperner’s Lemma

Le, Thanh and Le Van, Cuong and Pham, Ngoc-Sang and Sağlam, Çağrı

University of Wollongong, IPAG Business School, PSE, TIMAS, EM Normandie Business School, Métis Lab, Bilkent University,

Department of Economics

3 May 2020

Online at https://mpra.ub.uni-muenchen.de/103832/

MPRA Paper No. 103832, posted 03 Nov 2020 14:29 UTC

(2)

Direct Proofs of the Existence of Equilibrium, the Gale-Nikaido-Debreu Lemma and the Fixed Point

Theorems using Sperner’s Lemma

Thanh Le

Cuong Le Van

Ngoc-Sang Pham

C ¸ a˘grı Sa˘glam

§

October 28, 2020

Abstract

In this paper we use only Sperner’s lemma to prove the existence of general equi- librium for a competitive economy with production or with uncertainty and financial assets. We show that the direct use of Sperner’s lemma together with Carath´eodory’s convexity theorem and basic properties of topology such as partition of unit, finite covering of a compact set allow us to bypass the Kakutani fixed point theorem even in establishing the Gale-Nikaido-Debreu Lemma. We also provide a new proof of the Kakutani fixed point theorem based on Sperner’s lemma.

Keywords: Sperner lemma, Simplex, Subdivision, Fixed Point Theorem, Gale- Nikaido-Debreu Lemma, General Equilibrium.

JEL Classification: C60, C62, D5.

1 Introduction

The classic proofs of the existence of general equilibrium mainly rely on Brouwer and Kaku- tani fixed point theorems (Brouwer, 1911; Kakutani, 1941). They make use of either Gale- Nikaido-Debreu (Debreu,1959;Gale,1955;Nikaido,1956) or Gale and Mas-Colell (Gale and Mas-Colell, 1975, 1979) lemmas, the proofs of which in turn require Kakutani or Brouwer fixed point theorems.1

It is well known that the Sperner lemma (Sperner,1928) has historically formed the basis for these fixed point theorems. Sperner’s lemma is a combinatorial variant of the Brouwer fixed point theorem and actually equivalent to it.2 By enabling us to work with topological

University of Wollongong. Email address: thanhl@uow.edu.au

Corresponding author. IPAG Business School, Paris School of Economics, TIMAS. Email address:

levan@univ-paris1.fr

EM Normandie Business School, M´etis Lab. Email address: npham@em-normandie.fr

§Bilkent University, Department of Economics. Email address: csaglam@bilkent.edu.tr

1See, for excellent treatments of the existence of equilibrium,Debreu(1982) andFlorenzano(2003).

2For instance, Knaster, Kuratowski, and Mazurkiewicz (1929) use the Sperner lemma to prove the Knaster-Kuratowski-Mazurkiewicz lemma which implies the Brouwer fixed point theorem. Meanwhile,

(3)

spaces in a purely combinatorial way, it has proven to be useful in computing the fixed points of functions, critical points of dynamical systems, and the fair division problems (Su,1999).

However, this intuitive yet powerful lemma has not been fully exploited in the theory of general equilibrium. In particular, to what extent it allows us to dispense with the Kakutani fixed point theorem in proving the existence of general equilibrium remains to be explored further.

This paper highlights the role of the Sperner lemma as an alternative, purely conbinato- rial, non-fixed point theoretic approach to equilibrium analysis. To this end, we first prove the existence of general equilibrium for a competitive economy by using only the Sperner lemma without needing to recall neither the fixed point theorems nor the Gale-Nikaido- Debreu lemma. To ensure that this is not achieved at the price of generality, we consider both an economy with production and a two-period stochastic economy with incomplete financial markets.

The key point when applying the Sperner lemma is to construct a labeling which is proper (i.e., it satisfies Sperner condition) and, more importantly, will generate a point corresponding to an equilibrium price. In an earlier attempt, Scarf (1982) (page 1024) also uses the Sperner lemma to prove the existence of general equilibrium, but for a pure exchange economy. While the labeling of Scarf (1982) can be adapted for an economy with production, it is not easy to construct a labeling in a two-period economy with incomplete financial markets because the budget sets may have empty interiors when some prices are null. To overcome this difficulty, we introduce an artificial economy where all agents except for one have an additional income–ǫ–at the first period so that their budget sets have a non-empty interior for any prices system in the simplex. For this artificial economy, we can construct a proper labeling and hence prove the existence of equilibrium which depends on ǫ. Then, we let ǫ go to zero to get an equilibrium for the original economy.

Second, we use Sperner’s lemma to give a new proof of the Gale-Nikaido-Debreu lemma.

It is noteworthy that the existing proofs of the several versions of the Gale-Nikaido-Debreu lemma require the use of the fixed point theorems (see Florenzano (2009) for an excellent review). For instance, Debreu (1956, 1959) and Nikaido (1956) use the Kakutani fixed point theorem whileGale (1955) uses the Knaster-Kuratowski-Mazurkiewicz lemma. To the best of our knowledge, our paper is the first to present a proof of the Gale-Nikaido-Debreu lemma directly from Sperner’s lemma. More specifically, our proof relies on Sperner’s lemma, Carath´eodory (1907)’s convexity theorem, and the basic properties of topology such as the partition of unit and the finite covering of a compact set.

Last, but certainly not least, we provide a new proof of the Kakutani fixed point theorem by means of the Sperner lemma. By adapting the argument ofUzawa (1962) for continuous mapping, we also show that the Kakutani fixed point theorem can be obtained as a corollary of the Gale-Nikaido-Debreu lemma. Recall that Uzawa (1962) is only concerned with the equivalence between the Brouwer fixed point theorem and the Walras’ existence theorem.

There have been earlier attempts to use the Sperner lemma to prove the Kakutani fixed point theorem. For example, Sondjaja (2008) uses the Sperner lemma but she also requires

Yoseloff (1974) and Park and Jeong (2003) prove the Sperner lemma by using the Brouwer fixed point theorem. The reader is referred to Park (1999) for a more complete survey of fixed point theorems and Ben-El-Mechaiekh et al.(2009) for an excellent survey of general equilibrium and fixed point theory.

(4)

to make use of von Neumann (1937)’s approximation lemma. Shmalo (2018) proves the so- calledhyperplane labelinglemma, generalizing Sperner’s lemma, and uses it together with the approximate minimax theorem to prove the Kakutani fixed point theorem. In comparison, our method provides a more straightforward and direct proof of the theorem as it only uses the core notions of topology.

Note that the Sperner lemma and the mathematical tools that we have used to prove the existence of general equilibrium and the Gale-Nikaido-Debreu lemma dates back to 1928.

In this respect, our proofs suggest retrospectively that the existence of general equilibrium could have been proved almost two decades earlier before the seminal papers of Arrow and Debreu (1954) and Debreu (1959).3

The paper proceeds as follows. In Section 2, we review some basic concepts such as the notions of subsimplex, simplicial subdivision, Sperner’s lemma. In Section 3, we use the Sperner lemmma to prove the existence of general equilibrium (in two economies with either production or financial assets), the GND lemma as well as the Kakutani fixed point theorem.

Finally, Section 4concludes the paper.

2 Preliminaries

In this section, we introduce basic terminologies and necessary background for our work.

First, we present definitions from combinatorial topology based on which we state the Sperner lemma. After that, we provide a brief overview of correspondences and the maximum theo- rem which are extensively used for proving the existence of a general equilibrium.

2.1 On the Sperner lemma

Consider the Euclidean space Rn. Let e1 = (1,0,0, . . . ,0), e2 = (0,1,0, . . . ,0), . . ., and en = (0,0, . . . ,0,1) denote the n unit vectors of Rn. The unit-simplex ∆ of Rn is the convex hull of {e1, e2, . . . , en}. A simplex of ∆, denoted by [[x1, x2, . . . , xn]], is the convex hull of {x1, x2, . . . , xn} where xi ∈ ∆ for any i = 1, . . . , n, and the vectors (x1 −x2, x1 − x3, . . . , x1 −xn) are linearly independent, or equivalently, the vectors (x1, x2, . . . , xn) are affinely independent (i.e., if Pn

i=1λixi = 0 and Pn

i=1λi = 0 imply that λi = 0 ∀i).

Given a simplex [[x1, x2, . . . , xn]], a face of this simplex is the convex hull [[xi1, xi2, . . . , xim]]

with m < n, and {i1, i2, . . . , im} ⊂ {1,2, . . . , n}.

We now define the notions of simplicial subdivision (or triangulation) and labeling (see Border (1985) and Su (1999) for a general treatment) before stating the Sperner lemma.

Definition 1. T is a simplicial subdivision of ∆ if it is a finite collection of simplices and their faces ∆i, i= 1, . . . , p such that

• ∆ =∪pi=1i,

• ri(∆i)∩ri(∆j) =∅,∀i6=j.

3Recall that G´erard Debreu was awarded the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel in 1983 for having incorporated new analytical methods into economic theory and for his rigorous reformulation of the theory of general equilibrium.

(5)

Recall that if ∆i = [[xi1, xi2, . . . , xim]], then ri(∆i)≡ {x|x=Pm

k=1αkxk(i); P

kαk= 1; and

∀k :α(k)>0}.

Simplicial subdivision simply partitions an n-dimensional simplex into small simplices such that any two simplices are either disjoint or share a full face of a certain dimension.

Remark 1. For any positive integerK, there is a simplicial subdivisionTK ={∆K1 , . . . ,∆Kp(K)} of ∆ such that M esh(TK)≡maxi∈{1,...,p(K)}supx,y{kx−yk:x, y ∈∆Ki }<1/K. For exam- ple, we can take equilateral subdivisions or barycentric subdivisions.

We focus on the labeling of these subdivisions with certain restrictions.

Definition 2. Consider a simplicial subdivision of ∆. Let V denote the set of vertices of all the subsimplices of ∆. A labeling R is a function from V into {1,2, . . . , n}. A labeling R is said to be proper if it satisfies the Sperner condition:

x∈ ri[[ei1, ei2, . . . , eim]]⇒R(x)∈ {i1, i2, . . . , im}. In particular, R(ei) = i,∀i.

Note that the Sperner condition implies that all vertices of the simplex are labeled dis- tinctly. Moreover, the label of any vertex on the edge between the vertices of the original simplex matches with another label of these vertices. With these in mind, we can now state the Sperner lemma.

Lemma 1. (Sperner) Let T = {∆1, . . . ,∆p} be a simplicial subdivision of ∆. Let R be a labeling which satisfies the Sperner condition. Then there exists a subsimplex ∆i ∈T which is completely labeled, i.e. ∆i = [[x1(i), . . . , xn(i)]] with R(xl(i)) = l,∀l = 1, . . . , n.

The Sperner lemma guarantees the existence of a completely labeled subsimplex for any simplicially subdivided simplex in accordance with the Sperner condition. A proof of this lemma can be found in several text books or papers such as Sperner (1928), Berge (1959), Scarf and Hansen (1973),Le Van (1982). In particular, the original proof uses an inductive argument based on a complete enumeration of all completely labeled simplices for a series of lower dimensional problems. Meanwhile, proofs using constructive arguments date back toCohen (1967) and Kuhn(1968) (see Scarf(1982) for a demonstration of the constructive proof).

2.2 On correspondences

Let X ⊂ Rl, Y ⊂ Rm. A correspondence Γ from X into Y is a mapping from X into the set of subsets of Y. The graph of Γ is the set graphΓ = {(x, y) ∈ X ×Y : y ∈ Γ(x)}. A correspondence Γ :X →Y is closed if its graph is closed.

Definition 3. A correspondence Γ : X → Y is upper semicontinuous at point x if (i) Γ(x) is compact, non-empty, and (ii) for any sequence {xn} converging to x, for any sequence {yn} with yn∈Γ(xn),∀n, there exists a subsequence {ynk} which converges to y∈Γ(x).

Notice that ifX is compact then Γ is upper semicontinuous if and only if Γ is closed. It is also clear that if Γ is upper semicontinuous and K ⊂X is compact, then Γ(K) is compact.

Recall that if Γ is single valued, the notions of continuity, upper semicontinuity, and the lower semicontinuity turn out to be equivalent.

(6)

3 Main results

3.1 Using Sperner’s lemma to prove the existence of general equi- librium

We consider two hypothetical cases: an economy with production and an two-period stochas- tic economy with incomplete financial markets. Without recourse to the fixed-point theorems or the GND lemma, we are successful in establishing the results. Our proofs are novel as they only make use of the Sperner lemma and basic mathematical results.

3.1.1 Equilibrium existence in an economy with production

Consider an economy with L consumption goods, K input goods which may be capital or labor, I consumers, and J firms. Each consumerihas an initial endowment of consumption goodsωi ∈RL+, an initial endowment of inputsyi0 ∈RK+, and a utility function ui depending on her/his consumptions xi ∈ RL+. The firms produce consumption goods. Firm j has production functions Fj = (F1j, . . . , FLj) and uses a vector of inputs (y1j, . . . , yKj )∈RK+. The production functions satisfy Flj ≥0, and Fj 6= 0. We do not exclude that Flj = 0 for somel (e.g., firm j does not produce good l).

We adopt the following set of standard assumptions concerning the specifications of an economy with production.

Assumption 1. (i) Each utility function is strictly concave, continuous, and strictly in- creasing.

(ii) The endowments of consumption goods satisfy: ωi ≫0 (i.e., ω ∈RL++) ∀i.

(iii) The endowments of inputs satisfy: yj0 ≫0 (i.e., yj ∈RK++) ∀j.

(iv) For any l, Flj(0) = 0, and if Flj 6= 0 then it is strictly concave, strictly increasing.

(v) The firms distribute their profits among consumers. The share coefficientsθij,i= 1, . . . , I and j = 1, . . . , J are positive and satisfy P

iθij = 1,∀j.

In this economy, each firm j maximizes its profit given the prices p of outputs and the prices q of inputs. Let

Πj(p, q) = max

y∈RK+{p·Fj(y)−q·y}. We observe that for any (p, q), Πj(p, q)≥p·Fj(0)−q·0 = 0.

On the other hand, given the pricespof outputs and the pricesqof inputs, each consumer i solves the problem

maxui(xi) subject to xi ∈RL+ and p·xi ≤p·ωi+X

j

θijΠj(p, q) +q·yi0.

We now introduce the definitions of equilibrium and feasible allocation for such an economy with production.

Definition 4. An equilibrium is a list ((xi∗)i=1,...,I,(yj∗)j=1,...,J, p, q) satisfying (i) p ≫ 0, q ≫0, (ii) given prices, households and firms maximize their utility and profit respectively, (iii) all markets clear.

(7)

Definition 5. An allocation ((xi)i,(yj)j) is feasible if

(i) xi ∈RL+ for any i= 1, . . . , I, yj ∈RK+, for any j = 1, . . . , J, (ii) PI

i=1xi ≤PI

i=1ωi+PJ

j=1Fj(yj), (iii) PJ

j=1yj ≤PI i=1y0i.

The set of feasible allocations is denoted by F. It is convex and compact. We denote by Xi the set of allocations xi such that there exist (x−i)∈ (RL+)I−1 and (yj) which satisfy ((xi, x−i),(yj))∈ F. We denote by Yj the set of inputs (yj) such that there exist allocations (xi) which satisfy ((xi),(yj)) ∈ F. Note that all of these sets are convex, compact, and nonempty.

Let X be a closed ball of RL+ that contains all the Xi in its interior. Also, let Y be a closed ball of RK+ that contains all the sets Yj in its interior.

We will consider anintermediate economy in which the consumption sets equal toX and the inputs sets equal to Y. In this economy, given prices p and q, the behavior of each firm j can be recast as: maxyj∈Y{p·Fj(yj)−q·yj}. Accordingly, the behavior of each consumer i can be recast as

maxui(xi) subject to xi ∈X and p·xi ≤p·ωi+X

j

θijΠj(p, q) +q·y0i.

Definition 6. An equilibrium of the intermediate economy is a list((xi∗)i=1,...,I,(yj∗)j=1,...,J, p, q) that satisfies

(i) p ≫0, q ≫0,

(ii) For any i, xi∗ ∈X and p ·xi∗ =p·ωi+P

jθijΠj(p, q) +q ·y0i, (iii) For any i, xi ∈X, p·xi ≤p·ωi+P

jθijΠj(p, q) +q·y0i ⇒ui(xi)≤ui(xi∗), (iv) For any j, yj∗ ∈Y and Πj(p, q) =p·Fj(yj∗)−q ·yj∗,

(v) PI

i=1xi∗ =PI

i=1ωi+PJ

j=1Fj(yj∗) and PJ

j=1yj∗ =PI i=1y0i.

Since the utility functions and the production functions are strictly increasing, an equiv- alent definition can be reached by refining condition (v) in Definition 6. More precisely, an equilibrium in this intermediate economy is a list ((xi∗)i=1,...,I,(yj∗)j=1,...,J, p, q) that satisfies the conditions (i-iv) in Definition6 together with

(vi’) For any l = 1, . . . , L,PI

i=1xi∗l −PI

i=1ωil+PJ

j=1Flj(yj∗)

≤0, (vii’) For any k = 1, . . . , K,PJ

j=1ykj∗−PI

i=1y0,ki ≤0, (viii’) For any l = 1, . . . , L,pl PI

i=1xi∗l −PI

i=1ωli+PJ

j=1Flj(yj∗)

= 0, (viv’) For any k = 1, . . . , K, qkPJ

j=1ykj∗−PI

i=1y0,ki

= 0.

(8)

The following remark is important for the analysis of the equilibrium existence.

Remark 2. If (x, y) solves the problems of the consumers and the firms, then (x, y) satisfies Weak Walras Law:

p· X

i

(x∗i−ωi)−X

j

Fj(y)

+q· X

j

y∗j −X

i

y0i

≤0. (1)

However, if P

i(x∗i −ωi)−P

jFj(y) ≤ 0 and P

jy∗j −P

iyi0 ≤ 0, i.e., (x, y) ∈ F, since the utility functions are strictly increasing and the feasible set F is in the interior of X×Y, the allocation (x, y) satisfies Walras Law:

p· X

i

(x∗i−ωi)−X

j

Fj(y)

+q· X

j

y∗j −X

i

y0i

= 0. (2)

We now use the Sperner lemma to prove the existence of an equilibrium for the interme- diate economy. We will show that it is actually an equilibrium for the initial economy.

Proposition 1. Under above assumptions, there exists an equilibrium in the intermediate economy.

Proof. Letα >0.

Step 1. Consider the following transformed problem of the producer:

Πj,α(p, q) = max{p·Fj(yj)−q·yj :yj ∈Cj,α(p, q)}

whereCj,α(p, q) = {y∈Y :q·yj−p·Fj(yj)≤α}. Letηj,α(p, q) ={yj ∈Y :p·Fj(yj)−q·yj = Πj,α(p, q)}. Since the production function is strictly concave, ηj,αis a single-valued mapping.

We can directly prove, without using the Maximum Theorem (Berge, 1959), that ηj,α(p, q) is continuous in the set ∆ ≡ {(x1, . . . , xL+K) ≥ 0 : PL+K

i=1 xi = 1}. Indeed, let (p, q) ∈ ∆ and denote yj,α(p, q). We have thatp·Fj(y)−q·y ≥0>−α. Consider the sequence (pn, qn)∈∆ and converges to (p, q) when n tends to infinity. Let ynj,α(pn, qn).We have to prove thatyn converges to y. SinceCj,α(p, q) contains 0, we havep·Fj(y)−q·y ≥0.

Hence, for n large enough, we have pn·Fj(y)−qn·y >−α.

Again, by definition, we have Πj,α(pn, qn) =pn·Fj(yn)−qn·yn≥0>−α for any n.

When n→+∞, we can assume yn→y¯∈Y and hence, p·Fj(¯y)−q·y¯≥ −α. In other words ¯y ∈Cj,α(p, q).This implies

Πj,α(p, q) = p·Fj(y)−q·y ≥p·Fj(¯y)−q·y.¯ But since pn·Fj(y)−qn·y >−α, we have y ∈Cj,α(pn, qn). Therefore

Πj,α(pn, qn) =pn·Fj(yn)−qn·yn≥pn·Fj(y)−qn·y Letn →+∞. We get

p·Fj(¯y)−q·y¯≥p·Fj(y)−q·y

(9)

Therefore ¯y = y. We have proved that the mapping ηj,α is continuous. We then also get that the maximum profit Πj,α is a continuous function.

Step 2. Consider also the transformed problem of the consumer:

maxui(xi) subject to xi ∈X, p·xi ≤p·ωi+X

j

θijΠj,α(p, q) +q·y0i. It is easy to see that the setDi,α(p, q) ={xi :xi ∈X, p·xi ≤p·ωi+P

jθijΠj,α(p, q)+q·yi0}is convex and compact. Moreover, it has a non-empty interior. Indeed, observe that Πj,α(p, q)≥ 0. If p= 0 thenq > 0 and q·yi0 >0. We have 0 <P

jθijΠj,α(p, q) +q·yi0. If p6= 0, choose xi close toωi and xi ≪ωi. Thenp·(xi−ωi)<0≤P

jθijΠj,α(p, q) +q·yi0. For (p, q)∈∆ and i= 1, . . . , I, we define

ξα,i(p, q) = {xi ∈X :ui(xi)≥ui(x), if p·x ≤p·ωi+X

j

θijΠj,α(p, q) +q·y0i}. (3) The mapping ξα ≡ (ξα,i)Ii=1 is single-valued. We shall prove that ξα is continuous without using the Maximum Theorem (Berge, 1959).

Denote xi∗α,i(p, q), we have p·xi∗ ≤p·ωi+P

jθijΠj,α(p, q) +q·y0i.

Let (pn, qn) ∈ ∆ → (p, q) when n → +∞. Denote xi(n) = ξi(pn, qn). We can assume xi(n)→x¯i,α ∈X. Sincepn·xi(n)≤pn·ωi+P

jθijΠj,α(pn, qn) +qn·yi0, we have p·x¯i ≤p·ωi+X

j

θijΠj,α(p, q) +q·yi0 and hence ui(xi∗)≥ui(¯xi).

Let z ∈ intDi,α(p, q), i.e. it satisfies p·z < p·ωi+P

jθijΠj,α(p, q) +q·yi0. Then for n large enough,

pn·z < pn·ωi+X

j

θijΠj,α(pn, qn) +qn·yi0

This implies ui(xi(n))≥ ui(z) for any n large enough. Hence ui(¯xi)≥ ui(z). Actually this inequality holds for any z in the interior of Di,α(p, q). Take x0 ∈ intDi,α(p, q). For any integer m define zm = m1x0+ (1− m1)xi∗. Thenzm is in the interior of Di,α(p, q). We have

1

mui(x0) + (1− 1

m)ui(xi∗)≤ui(zm)≤ui(¯xi)

Let m → +∞. We get ui(xi∗) ≤ ui(¯xi). Hence ¯xi = xi∗. We have prove that ξα,i is continuous.

Step 3. DenoteN =L+K andπ = (p, q)∈∆, and define the excess demand mappings ξα(π) =

XI i=1

α,i(π)−ωi)− XJ

j=1

Fjj,α(π))

ηα(π) = XJ

j=1

ηj,α(π)− XI

i=1

yi0 ζ(π) = (ξα(π), ηα(π)).

(10)

According to Steps 1 and 2, the mapping ζ is continuous.

Step 4. We will use the Sperner lemma to prove that there exists π ∈ ∆ such that ζj)≤ 0 ∀j. Indeed, let K >0 be an integer and consider a simplicial subdivision TK of

∆ such that M esh(TK) < 1/K and define a labeling R as follows: For π ∈ ∆, R(π) = i where isatisfies ζi(π) ≤ 0. We can see that the labeling R is well-defined (because of Weak Walras Law) and satisfies Sperner condition.4 According to the Sperner lemma, there exists a completely labeled subsimplex [[¯πK,1,π¯K,2, . . . ,π¯K,n]] such that R(¯πK,j) = j, i.e., ζj(¯πK,j) ≤ 0, ∀j = 1, . . . , N. Let K go to +∞, the vertices {π¯K,j} converge to the same point π ∈∆. This point satisfies ζj)≤0 ∀j.

Step 5. From Remark 2, Walras Law holds. Hence, P

jπjζj) = 0 and we have actually πjζj) = 0,∀j.

Finally, we claim that Πj,α(p, q) = max{p·Fj(yj)−q ·yj :yj ∈Y}.Indeed, if there exists y ∈ Y such that p·Fj(y)−q ·y > Πj,α(p, q) ≥ 0, then q ·y−pFj(y) <0 < α and that is a contradiction.

Condition Πj,α(p, q) = max{p·Fj(yj)−q·yj :yj ∈Y}and the definition of ξα,i(p, q) imply the optimality of consumers’ allocation.

We have proved that there exists an equilibrium in the intermediate economy.

The following proposition allows us to move from an equilibrium in the intermediate economy to an equilibrium in the initial economy.

Proposition 2. ((xi∗)i=1,...,I,(yj∗)j=1,...,J, p, q) is an equilibrium for the initial economy.

Proof. First observe that if there exists y∈RK+ such that

p·Fj(y)−q·y > p·Fj(y)−q·y = Πj,α(p, q)≥0

then q ·y−pFj(y)<0< α and that is a contradiction. By consequence, we get that p·Fj(y)−q·y = Πj(p, q) = max{p·Fj(yj)−q·yj :yj ∈RK+}.

Now fix some i and take x ∈RL+ satisfying ui(x) > ui(xi∗). We have to prove that p ·x >

p ·ωi +P

jθijΠj(p, q) +q·yi0. Of course, this is the case if x ∈ X. We now consider the case where x /∈ X. Since xi∗ is in the interior of X, there exists λ ∈ (0,1) such that λx+ (1−λ)xi∗ ∈X. We haveui(λx+ (1−λ)xi∗)≥λui(x) + (1−λ)ui(xi∗)> ui(xi∗).Hence, we have

p·(λx+ (1−λ)xi∗)> p·ωi+X

j

θijΠj(p, q) +q·yi0 =p·xi∗

⇔λp·x > λp·xi∗ ⇔p·x > p·xi∗ =p·ωi+X

j

θijΠj(p, q) +q·y0i.

Remark 3. It is interesting to note that our proof of the existence of general equilibrium requires only the Sperner lemma and elementary mathematical results which were available before 1930. We do not need to use the Maximum Theorem proven by Berge (1959).

4This labeling is similar to that inScarf(1982), page 1024.

(11)

3.1.2 Equilibrium existence in an economy with financial assets

In this section, we use the Sperner lemma to prove the existence of an equilibrium in a two-period stochastic economy with incomplete financial markets. We briefly present here some essential notions. For a full exposition, see Magill and Quinzii (1996) and Florenzano (1999).

Consider an economy with two periods (t = 0 and t = 1), L consumption goods, J financial assets, andI agents. There is no uncertainty in period 0 while there areS possible states of nature in period 1. In period 0, each agent i ≤ I consumes and purchases assets.

The consumption prices are denoted by p0 ∈ RL+ in the first period, ps ∈ RL+ in the state s of period 1. Let π ≡ (p0, p1, . . . , pS). Each consumer has endowments of consumption good ω0i ∈ RL+ in period 0 and ωsi ∈ RL+ in state s of period 1. Any agent i has a utility functionUi(xi0, xi1, . . . xiS) wherexisis her consumption at states. There is a matrix of returns depending onπ of financial assets which is the same for any agent. Typically, if agenti≤I purchases zi quantity of assets in period 0, in period 1, at state s, she/he will obtain an income (positive or negative) PJ

j=1Rs,j(π)zj. The returns R(π) can be represented by a matrix

R=







R1,1(π)R1,2(π) . . . R1,J(π) R2,1(π)R2,2(π) . . . R2,J(π)

. . . . . . . . . . . . . . .

RS,1(π)RS,2(π) . . . RS,J(π)







We denote by Rs(π) = (Rs,1(π), Rs,2(π), . . . , Rs,J(π)) the sth row of R(π). Typically, the constraints faced by agent iare

p0·(xi0−ω0i) +q·zi ≤0,

ps·(xis−ωsi)≤Rs(π)·zi ∀s= 1, . . . , S.

We make use of the following set of standard assumptions.

Assumption 2. (i) For any i= 1, . . . , I, the consumption set is RL+ the assets set Zi =RJ. (ii) For any i= 1, . . . , I, ω0i ∈RL++, ωsi ∈RL+ for any states in period 1.

(iii) Rs,j(π)>0, for any s, any j, any π.

(iv) rank R(π) = J, for any π and the map π→R(π) is continuous.

(v) For any i= 1, . . . , I, Ui is strictly increasing, continuous, and strictly concave.

We now introduce the definitions of complete and incomplete asset markets, feasible allocations, and the notion of equilibrium in an economy with financial assets.

Definition 7. The assets market is called complete if S =J and incomplete if S > J. Definition 8. An equilibrium of this economy is a list xi∗, zi∗)Ii=1, xI+1∗,(p, q)

where (xi∗, zi∗)Ii=1 ∈(Xi)I ×(Zi)J, (p, q)∈RL++×RJ++ such that

(12)

(i) For any i, (xi∗, zi∗) solve the problem

maxUi(xi0, xi1, . . . , xiS)

subject to: p0·(xi0−ω0i) +q·zi ≤0 (4a) ps·(xis−ωis)≤Rs(π)·zi, s= 1, . . . , S (4b) (ii) PI

i=1(x∗is −ωsi) = 0 for any s= 0,1, . . . , S and PI

i=1z∗i = 0.

Definition 9. The allocations ((xi, zi)i)∈(Xi)I ×(ZI)I are feasible if (i) PI

i=1(xi −ωi) ≤ 0 and (ii) PI

i=1zi = 0. Accordingly, take α > 0 and define the sets Fc = {(xi)i ∈ (Xi)I : PI

i=1(xi − ωi) ≤ α} and Ff = {(zi)i ∈ (Zi)I : P

izji = 0,∀j}. Moreover, denote the projection of Fc on Xi by Xbi.

The following lemma will be useful in proving the existence of equilibrium.

Lemma 2. Let (zi)∈RJ×I satisfy that: for all i, there exists (xi)∈Fc such that

∀s= 1, . . . , S, ps·(xis−ωsi) =Rs(π)·zi where kpsk ≤1,∀s. Then there exists β >0 such that kzik ≤β,∀i.

Proof. Assume that there exists a sequence (zi(n))n with kzi(n)k → +∞ when n → +∞. We have, for any n, ∀s = 1, . . . , S, ps(n)·(xis(n)−ωsi) = Rs(π(n))·zi(n). We can assume that π(n) → π ∈ ∆. We obtain that, ∀s = 1, . . . , S,ps(n)·(xkziis(n)k(n)−ωis) = Rs(π(n))· kzzii(n)(n)k. We can suppose kzzii(n)(n)k →ζ 6= 0. Let n→+∞. We get 0 =Rs(π)·ζ. Since rank R(π) = J, we haveζ = 0: a contradiction.

Let Bc be a ball of RL, centered at the origin, which contains any ˆXi in its interior. Let us consider an intermediate economy in which the consumption set is Xei =Bc for any i.

Definition 10. An equilibrium of this intermediate economy is a list (xi∗, zi∗)Ii=1,(p, q) where (xi∗, zi∗)Ii=1 ∈(Xei)I ×(Zei)J, (p, q)∈RL++×RJ++ such that

(i) For any i, xi∗ solve the problem

maxUi(xi0, xi1, . . . , xiS) (5a) subject to: ∃zi ∈RJ, p0·(xi0−ωi0) +q·zi ≤0, (5b) ps·(xis−ωis)≤Rs(π)·zi, s= 1, . . . , S (5c) xi ∈Xei ∀s= 0,1, . . . , S. (5d) (ii) PI

i=1(x∗is −ωsi) = 0 for any s= 0,1, . . . , S and PI

i=1z∗i = 0.

We aim to provide a new proof (by using the Sperner lemma) of the following result which corresponds to Theorem 1 inCass (2006) or Theorem 7.1 in Florenzano(1999). Notice that our proof does not require that the returns are nominal asCass(2006) andFlorenzano(1999) did. Our proof works for nominal, and num´eraire assets as well by choosing a num´eraire which is strictly positive. Actually, it works for any returns matrixR(p) which is continuous and whose rank equals J which ensures that the feasible set is bounded (see Lemma 2).

(13)

Proposition 3. Under above assumptions, there exists an equilibrium (xi∗, zi∗)Ii=1,(p, q) with q =PS

s=1Rs(π).

Proof. Observe that, by using the same argument in the proof of Proposition 2 in Section 3.1.1, we can prove that an equilibrium of the intermediate economy is indeed an equilibrium for the initial economy. As such, it remains to prove the existence of equilibrium in the intermediate economy. To do so, we proceed in two steps. First, we use the Sperner lemma to prove that there exists actually aCass equilibrium. Second, we show that this equilibrium constitutes an equilibrium of the intermediate economy.

We now define and prove the existence of Cass equilibrium.

Definition 11. Cass equilibrium is a list (¯xi)Ii=1,(¯zi)Ii=2,(¯p,q)¯

such that (¯xi)Ii=1,(¯zi)Ii=2 ∈ (Bc)I ×(Bf)I−1, (¯p,q))¯ ∈RL++×RJ++, and π¯ = (¯p,q)¯ where

(i) x¯1 solves the consumer 1 problem under the constraint x1 ∈Bc, π¯·(x1−ω1)≤0.

(ii) For i= 2, . . . , I, x¯i solves the consumer i’s problem

maxUi(xi0, xi1, . . . , xiS) subject to: ∃zi ∈RJ, p¯0·(xi0−ω0i) + ¯q·zi ≤0,

¯

ps·(xis−ωis)≤Rs(π)·zi ∀s ≥1 xi ∈Bc ∀i.

(iii) q¯=P

sRs(π) and PI

i=1(¯xi−ωi) = 0.

Lemma 3. There exists a Cass equilibrium.

Proof. Let π = (p0, p1, . . . , pS) ∈ ∆ where ∆ denotes the unit-simplex of RL(S+1). Let ǫ be such that 0< ǫ < (I−1)α .

Agent 1 solves the following problem

maxU1(x1) subject to x1 ∈Xe1, π·(x1 −ω1)≤0.

Any agent i (i≥2) solves the following problem maxUi(xi) subject to: xi ∈Xei, zi ∈Zei,

∃zi ∈RJ, p0·(xi0−ωi0) + (X

s

Rs(π))·zi ≤ǫ, ps·(xis−ωsi)≤Rs(π)·zi ∀s≥1.

The budget set of agent 1 has a nonempty interior since π ∈ ∆. To prove the budget sets of the agents i ≥ 2 have nonempty interiors, we observe that xis = ωsi, s = 0,1, . . . , S and zi > 0 such that P

sRs(π)zi < ǫ are in the interior of these budget sets. Therefore, the optimal values (x∗1, x∗2ǫ . . . , x∗Iǫ ) and (zǫ∗2, . . . , zǫ∗I) are continuous mappings with respect to π.5 For any π, we have

π· XI

i=1

(x∗i(π)−ωi)≤(I−1)ǫ.

5We can prove this continuity by applying the Maximum Theorem (Berge,1959) or adapting our argument in Step 2 of the proof of Proposition1.

(14)

Define the excess demand mapping ξ by ξ(π) =

XI i=1

(x∗i(π)−ωi).

It is obvious that ∀π ∈∆, π·ξ(π)≤(I−1)ǫ.

(Using the Sperner lemma) Denote N = (S+ 1)L. Let K > 0 be an integer and consider a simplicial subdivision TK of the unit-simplex ∆ of RN such that M esh(TK) <

1/K. We define the following labeling r. For any π ∈∆, r(π) =t if ξt(π) ≤(I−1)ǫ. Such a labeling is well defined. Moreover, it satisfies Sperner condition. Indeed, we see that:

• For t ∈ {1, . . . , N}. If π = et (recall that et is a unit-vector of RN), then (I −1)ǫ ≥ et·ξ(et) = ξt(et). We label r(et) =t.

• If π ∈ [[ei1, . . . , eim]] with m < N, then (I −1)ǫ ≥ π· ξ(π) = P

q∈{i1,...,im}πqξj(π).

There must existsq ∈ {i1, . . . , im}with ξq(π)≤(I−1)ǫ. We labelr(π) = qwith some q∈ {i1, . . . , im}.

So, the labelingrsatisfies Sperner condition. According to the Sperner lemma, there exists a completely labeled subsimplex [[¯π1(K), . . . ,π¯N(K)]], i.e.,ξt(¯πt(K))≤(I−1)ǫ∀t= 1, . . . , N. Observe that

∀t= 1, . . . , N, XI

i=1

x∗i(¯πt(K))−ωi

≤(I −1)ǫ < α. (6) Let K → +∞. Then, for any t ∈ {1, . . . , N}, ¯πt(K) → π(ǫ) ∈ ∆. We have ξq(ǫ)) ≤ (I−1)ǫ < α, for all q. It follows from (6) that

XI i=1

x∗i(ǫ))−ωi

≤(I−1)ǫ < α. (7)

Write π(ǫ) = (p0(ǫ), p1(ǫ), . . . , pS(ǫ)). Because of (7) and the fact that utility functions are strictly increasing, we obtain

π(ǫ)·(x∗1(ǫ))−ω1) = 0 (8) that implies π(ǫ)≫0. Hence, for any i≥2,

p0(ǫ)·(x∗i0(ǫ))−ωsi) + (X

s

Rs(ǫ))z∗i(ǫ)) =ǫ, ps(x∗i(ǫ))−ωsi) =Rs(ǫ))·z∗i(ǫ)), s= 1, . . . , S.

From Lemma 2, we have kz∗i(ǫ))k ≤β.

Let ǫ→0, we have that

• π(ǫ)→π,¯

• x∗1(ǫ))→x¯1 =x∗1(¯π)⇒π¯≫0,

(15)

• π¯ ≫ 0 ⇒ ∀i ≥ 2, x∗i(ǫ)) → x¯i = x∗i(¯π), z∗i(ǫ)) → z¯i = z∗i(¯π), i.e., for i ≥ 2, (¯xi,z¯i) solves the problem of agenti for given prices ¯π.

Note from (7) thatPI

i=1(¯xi−ωi)≤0 and from (8) that ¯π·(PI

i=1(¯xi−ωi) = 0⇒π¯pP

i(¯xip− ωpi) = 0, p= 1, . . . , N.

Since ¯π≫0, we deduce thatPI

i=1(¯xip−ωpi) = 0,∀p= 1, . . . , N, or equivalentlyPI

i=1(¯xi− ωi) = 0. We have proved the existence of aCass equilibrium.

We move from Cass equilibrium to an equilibrium in the intermediate economy.

Lemma 4. There exists an equilibrium in the intermediate economy with q¯=P

sRs(π).

Proof. Since PI

i=1(¯xis−ωsi) = 0 ∀s≥1, we get that

∀s≥1,0 = ¯ps· XI

i=1

(¯xis−ωsi) = ¯ps·(¯x1s−ωsi) + ¯ps· XI

i=2

(¯xis−ωsi).

Denote ¯z1 =−P

i≥2i. We have ¯ps·PI

i=2(¯xis−ωsi) = Rs(¯π)·z¯1 which implies that X

s≥1

¯

ps·(¯x1s−ωsi) = X

s

Rs(¯π)

·z¯1 = ¯q·z¯1. By combining this with the fact that ¯p0 ·(¯x10 −ω01) +P

s≥1s·(¯x1s−ω1s) = 0, we get that

¯

p0·(¯x10−ω10) + ¯q·z¯1 = 0.

It is easy to prove that ¯x1 solves the problem (5a-5d).

Remark 4 (equilibrium price versus no-arbitrage price). Our above proof of the existence of competitive equilibrium leads to a conclusion that: an equilibrium exists if and only if there exists a no-arbitrage assets price. Indeed, any no-arbitrage price is the strictly positive convex combination of financial returns. Accordingly, take a no-arbitrage price. Using the Cass trick we obtain an equilibrium. Conversely, for any financial equilibrium, under the assumption that the utility functions are strictly increasing, the first order conditions show that an equilibrium asset price is a no-arbitrage price.

Remark 5. When we use the utility functions and production functions, we can skip the use of the Kakutani fixed point theorem. This theorem is required when the utility functions or the production functions are not strictly concave, or instead of utility functions and production functions we have preference orders for the consumers and production sets. In these cases, the demands of the consumers or of the firms are not necessarily single valued. They are upper semicontinuous correspondences with convex compact values. However, if the utility functions and the production are only concave, we can approximate them by a family of strictly concave utility functions and production functions as follows

For ε >0, define uiε(x) =ui(x) +εv(x), Fεj(k) = Fj(k) +εG(k) where ε >0, v and G are strictly concave.

For any ε >0 we get an equilibrium (xi∗(ε))i=1,...,I,(yj∗(ε))j=1,...,J, p(ε), q(ε)

. Let ε go to zero. It is easy to prove that the limit of this list constitutes an equilibrium for the initial economy.

(16)

3.2 Using Sperner’s lemma to prove the Gale-Nikaido-Debreu lemma

The customary proofs of the existence of a general equilibrium also make use of either the GND lemma (Debreu, 1956, 1959; Gale, 1955; Nikaido, 1956) or the Gale and Mas-Colell lemma (Gale and Mas-Colell, 1975, 1979) whose proofs, in turn, require the Kakutani fixed point theorem or the Knaster-Kuratowski-Mazurkiewicz lemma. (See Florenzano (1982), Florenzano and Le Van (1986) for more detailed discussions.) In what follows, we use the Sperner lemma and well-known mathematical results to prove several versions of the GND lemma.

Lemma 5 (Gale-Nikaido-Debreu lemma). Let ∆ be the unit-simplex of RN. Let ζ be an upper semi-continuous correspondence with non-empty, compact, convex values from ∆ into RN. Suppose ζ satisfies the following condition:

∀p∈∆, ∀z ∈ζ(p), p·z ≤0. (9)

Then there exists p¯∈∆ such that ζ(¯p)∩RN 6=∅. Proof. LetA = max{kzk1 :z ∈ζ(∆)}.

Step 0. Let ǫ ∈ (0,1). Since ∆ is compact, there exists a finite covering of ∆ with a finite family of open balls B˜(xi(ǫ), ǫ)

i=1,...,I(ǫ). Take a partition of unity subordinate to the family B˜(xi(ǫ), ǫ)

i=1,...,I(ǫ), i.e. a family of continuous non negative real functions (αi)i=1,...,I(ǫ)from ∆ inR+such that Suppαi ⊂B(xi(ǫ), ǫ),∀iandPI(ǫ)

i=1αi(x) = 1,∀x∈∆.6 Takeyi(ǫ)∈ζ(xi(ǫ)) ∀i.

Step 1. We define the function fǫ : ∆→ ∆ by fǫ(x) = PI(ǫ)

i=1αi(x)yi(ǫ). This function is continuous.

Step 2. We claim that: x · fǫ(x) ≤ ǫA ∀x ∈ ∆. Let x ∈ ∆, there exists a set J(x) ⊂ {1, . . . , I(ǫ)} such that x ∈ ∩i∈J(x)B˜(xi(ǫ), ǫ). We have fǫ(x) = P

i∈J(x)αi(x)yi(ǫ) with P

i∈J(x)αi(x) = 1. We have

∀i∈J(x), xi(ǫ) = x+ǫui(x), with some ui(x)∈B(0,1) which implies that: ∀i ∈ J(x), yi(ǫ) ∈ ζ(xi(ǫ)) = ζ(x+ǫui(x)) ⊂ ζ B(x, ǫ)

. By conse- quence, fǫ(x)∈ co

ζ B(x, ǫ)

. According to Carath´eodory’s convexity theorem,7 we have a decomposition

fǫ(x) =

NX+1 i=1

βi(x, ǫ)˜yi(x, ǫ) with ˜yi(x, ǫ)∈ζ(x+ǫui) whereui ∈B(0,1), βi(x, ǫ)≥0, PN+1

i=1 βi(x, ǫ) = 1. From this, we

6For the notion of partition of unity, see, for instance,Aliprantis and Border(2006)’s Section 2.19.

7Carath´eodory(1907)’s convexity Theorem states that: In an n-dimensional vector space, every vector in the convex hull of a nonempty set can be written as a convex combination using no more thann+ 1 vectors from the set. For a simple proof, see Florenzano and Le Van (2001)’s Proposition 1.1.2 orAliprantis and Border(2006)’s Theorem 5.32.

Referenzen

ÄHNLICHE DOKUMENTE

The tradition of general equilibrium models and general equilibrium theory is based on the paradigm of a perfectly competitive economy, with prices determined by market forces

It follows from this model that under equilibrium conditions the total value of goods of some branch consumed in other branches, equals the total value of goods of

Otra característica del enfoque de Arrow y que tiene su origen en la filosofía utilitarista es el consecuencialismo; esto significa que toda acción, regla o institución es juzgada

The unit supply bundles of all agents constitute the n-by- m unit supply matrix B ( ) ε. In equilibrium the unit expenditure of each agent is no less than its unit revenue [6-8].

Typically, a SAM in an open economy contains five types of accounts: 1) activities of accounts (or production accounts), 2) products and services accounts (products accounts),

In this work we are going to use the excess demand approach to prove the existence of equilibrium [2, Section 3]: we just impose conditions a function should possess to be the

For example, it is well-known that WARP holds for the excess demand function of an exchange economy in which endow- ments are collinear and all agents have demand functions obeying

Proposition 1: The conflict between the partial and general equilibrium vanishes in the described production economy with decreasing returns under sufficiently small val- ues of